| L(s) = 1 | + 3-s + 7-s + 9-s + 2·13-s − 3·17-s − 3·19-s + 21-s + 23-s + 27-s − 6·29-s − 2·31-s + 3·37-s + 2·39-s − 3·41-s − 12·43-s − 47-s − 6·49-s − 3·51-s + 6·53-s − 3·57-s + 3·59-s + 10·61-s + 63-s − 6·67-s + 69-s + 7·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 0.727·17-s − 0.688·19-s + 0.218·21-s + 0.208·23-s + 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.493·37-s + 0.320·39-s − 0.468·41-s − 1.82·43-s − 0.145·47-s − 6/7·49-s − 0.420·51-s + 0.824·53-s − 0.397·57-s + 0.390·59-s + 1.28·61-s + 0.125·63-s − 0.733·67-s + 0.120·69-s + 0.830·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.51008199473410, −13.22351255404244, −12.82266306543511, −12.23321981693188, −11.61639249458732, −11.08176492127781, −10.97492586827834, −10.09014162134957, −9.848350993513805, −9.167902341060866, −8.663989244826887, −8.409568330871580, −7.837550939166400, −7.311445769561716, −6.633983501114712, −6.447703391731618, −5.554849674902248, −5.160628180130830, −4.479767346098027, −3.983446722082040, −3.465912761577280, −2.880478107323682, −1.960273330784814, −1.884518138015969, −0.8985916222084224, 0,
0.8985916222084224, 1.884518138015969, 1.960273330784814, 2.880478107323682, 3.465912761577280, 3.983446722082040, 4.479767346098027, 5.160628180130830, 5.554849674902248, 6.447703391731618, 6.633983501114712, 7.311445769561716, 7.837550939166400, 8.409568330871580, 8.663989244826887, 9.167902341060866, 9.848350993513805, 10.09014162134957, 10.97492586827834, 11.08176492127781, 11.61639249458732, 12.23321981693188, 12.82266306543511, 13.22351255404244, 13.51008199473410
